Stability for socle-projective categories of type $\mathbb{A}$

TL;DR

The paper extends stability concepts to the non-abelian category of socle-projective poset representations $ ext{P-spr}$ for posets of type $ ext{A}$. It proves both bilinear-form and geometric-model-based stability: indecomposables are stable with a bilinear-form weight $b_{ ext{P}}( ext{dim} extbf{U}, ext{dim} extbf{W})-b_{ ext{P}}( ext{dim} extbf{W}, ext{dim} extbf{U})$, and also via a central-charge stability arising from BGMS-inspired polygon models. The authors construct a new geometric realization of $ ext{mod}_{sp}k ext{P}$ through sp-segments and establish a categorical equivalence with socle-projective modules; this links algebraic stability with geometric objects and AR-structure. The work unifies two viewpoints—bilinear-form stability and geometric central-charge stability—and provides explicit coordinates and translations between the geometric model and the representation category, broadening moduli-space perspectives for poset representations of type $ ext{A}$.

Abstract

We extend the notion of stability in the non-abelian category of poset representations (introduced by Futorny and Iusenko) to the category of socle-projective representations of a given $r$-peak poset $¶$. When $¶$ is a poset of type $\mathbb{A}$, we demonstrate in two distinct ways that every indecomposable peak $¶$-space is stable. First, this is shown using a bilinear form associated with the poset. Second, we prove it by observing that a stability function derived from a geometric model ensures that all indecomposable objects are stable. Along the way, we provide a new geometric realization of the category of socle-projective representations, inspired by the work of Schiffler and Serna [\textit{J. Pure Appl. Algebra} \textbf{224} (2020), no.~12, 106436, 23 pp.; MR4101480]. Finally, we establish a connection between the geometric perspective and the bilinear form approach.

Figures (9)

• Figure 1: Polygon $P(Q)$ given by Example \ref{['alien set']}
• Figure 2: Set of all suitable line segments in $P(Q)$.
• Figure 3: Configuration of suitable line segments.
• Figure 4: Diagram of the mesh relations in $\mathcal{C}_{P_{\text{sp}}(Q^F)}$
• Figure 5: Auslander-Reiter quiver of $\mathop{\mathrm{mod}}\nolimits_{sp}(\Bbbk \mathcal{P})$

Theorems & Definitions (61)

• Proposition 1.1
• proof
• Proposition 1.2
• proof
• Proposition 1.3
• proof
• Lemma 1.4
• proof
• Proposition 1.5
• Proposition 1.6: Harder-Narasimhan filtration